To determine which of the given reactions is a single-replacement reaction, we need to identify the type of reaction for each option.
A single-replacement reaction (also known as a single-displacement reaction) involves one element being displaced from a compound by another element. In general form, it can be represented as:
[tex]\[ A + BC \rightarrow AC + B \][/tex]
where element A replaces element B in compound BC.
Let's analyze each given reaction:
### Reaction A:
[tex]\[ CH_4 + 2 O_2 \rightarrow CO_2 + 2 H_2O \][/tex]
This is a combustion reaction where methane ([tex]\( CH_4 \)[/tex]) burns in the presence of oxygen ([tex]\( O_2 \)[/tex]), producing carbon dioxide ([tex]\( CO_2 \)[/tex]) and water ([tex]\( H_2O \)[/tex]). Combustion reactions typically involve a hydrocarbon reacting with oxygen. This is not a single-replacement reaction.
### Reaction B:
[tex]\[ 2 AgNO_3 + Cu \rightarrow 2 Ag + Cu(NO_3)_2 \][/tex]
Here, we see copper ([tex]\( Cu \)[/tex]) reacting with silver nitrate ([tex]\( AgNO_3 \)[/tex]). Copper displaces silver from the nitrate compound, forming copper(II) nitrate ([tex]\( Cu(NO_3)_2 \)[/tex]) and elemental silver ([tex]\( Ag \)[/tex]). This fits the pattern of a single-replacement reaction, as one element (Cu) replaces another element (Ag) in a compound.
### Reaction C:
[tex]\[ CaCl_2 + Na_2CO_3 \rightarrow CaCO_3 + 2 NaCl \][/tex]
In this reaction, calcium chloride ([tex]\( CaCl_2 \)[/tex]) reacts with sodium carbonate ([tex]\( Na_2CO_3 \)[/tex]) to produce calcium carbonate ([tex]\( CaCO_3 \)[/tex]) and sodium chloride ([tex]\( NaCl \)[/tex]). This is a double-replacement reaction (also known as a double-displacement reaction) because the cations and anions of the two reacting compounds exchange partners. It is not a single-replacement reaction.
### Reaction D:
[tex]\[ H_2 + Cl_2 \rightarrow 2 HCl \][/tex]
This reaction involves hydrogen gas ([tex]\( H_2 \)[/tex]) reacting with chlorine gas ([tex]\( Cl_2 \)[/tex]) to form hydrogen chloride ([tex]\( HCl \)[/tex]). This is a synthesis reaction, where two elements combine to form a single compound. It is not a single-replacement reaction.
Based on the analysis, the single-replacement reaction is:
[tex]\[ \boxed{B} \][/tex]
